I personally have an enclosure that is a touch bigger than you have suggested 36x24x12 I think you size is fine but I'd suggest 16 tall. The biggest reason is it is much easier to clean. It is heavy and awkward enough that caring it to a sink to clean is a big job so it is a reach inside and the little extra height is easier to deal with. The doors are taller, my 12 inch one has 3 inch lip at the bottom and 2 at the top and that makes the viewable area only 7 inch it is quite small.

Heating through wood will work but it is very inefficient. I suggest that replace the wood bottom with plastic HDPE or PVCx works well both are easily workable and heat ok. (1/4 inch) just use a bit heavier side and screw up through the bottom. That would be my suggestions.

I hope it helps spark some ideas.
Alex